Question

select the correct answer.
the points d(-2, 2), e(2, 5), and f(5, 1) form △def in the coordinate plane. what condition verifies that the triangle is a right triangle?
a. two of the sides of the triangle are equal.
b. the product of the slopes of two of the sides is 1.
c. three of the sides of the triangle are equal.
d. the product of the slopes of two of the sides is -1.
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Explanation:

Step1: Calculate the slope of \(DE\)

The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). For points \(D(-2,2)\) and \(E(2,5)\), \(m_{DE}=\frac{5 - 2}{2-(-2)}=\frac{3}{4}\)

Step2: Calculate the slope of \(EF\)

For points \(E(2,5)\) and \(F(5,1)\), \(m_{EF}=\frac{1 - 5}{5 - 2}=\frac{-4}{3}\)

Step3: Calculate the slope of \(FD\)

For points \(F(5,1)\) and \(D(-2,2)\), \(m_{FD}=\frac{2 - 1}{-2 - 5}=\frac{1}{-7}=-\frac{1}{7}\)

Step4: Check the product of slopes

We know that if two lines are perpendicular, the product of their slopes is \(- 1\). Here, \(m_{DE}\times m_{EF}=\frac{3}{4}\times\frac{-4}{3}=-1\)

Answer:

D. The product of the slopes of two of the sides is \(-1\)